On the asymptotic behavior of the Diaconis-Freedman chain on [0,1]

We describe the asymptotic behavior of the Diaconis-Freedman chain on [0,1], using technics developed to study iterated Lipschitz functions systems with possibly place dependent probabilities. Under some general conditions on this family of probabilities and using quasi-compact linear operators technics, we obtain a necessary and sufficient condition for the uniqueness of the stationary probability measure for this chain and explore the case when it does not hold.